In the projective space $\mathrm{PG}(3,q)$, we consider the orbits of lines under the stabilizer group of the twisted cubic. It is well known that the lines can be partitioned into classes every of which is a union of line orbits. All types of lines forming a unique orbit are found. For the rest of the line types (apart from one of them) it is proved that they form exactly two or three orbits; sizes and structures of these orbits are determined. Problems remaining open for one type of lines are formulated. For $5\le q\le37$ and $q=64$, they are solved.